a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and...
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and let A be an n x n skew-symmetric matrix. Simplify A(A^2(BA)^-1)^T
b. Let A be a square matrix such that A^3 = 0. A is then called a nilpotent matrix. Define another matrix B by the expression B = I - A; Show that B is invertible and that its inverse is I + A + A^2
c. Let B = (-2,0,0 ; 0,0,0 ; 0,0,3) Find B^100 and e^B
Homework Answers
Help on Questions 1-3 Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x...
Help on Questions 1-3
Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P su...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
[1 2 37 1. Is the matrix 1 0 1 invertible? If yes, what is its...
[1 2 37 1. Is the matrix 1 0 1 invertible? If yes, what is its inverse? [O 2 -1 2. A matrix is called symmetric if At = A. What can you say about the shape of a symmetric matrix? Give an example of a symmetric matrix that is not a zero matrix. 3. A matrix is called anti-symmetric if A= -A. What can you say about the shape of an anti- symmetric matrix? Give an example of an...
7. Claim: Let A be an (n × n) (square) matrix. ·Claim: If A s invertible and AT = A-1 , then the columns of A form an orthonormal basis for R . Claim: If the columns of A form an orthogonal basis...
7. Claim: Let A be an (n × n) (square) matrix. ·Claim: If A s invertible and AT = A-1 , then the columns of A form an orthonormal basis for R . Claim: If the columns of A form an orthogonal basis for Rn, then A is invertible and A A-1 . Claim: If the columns of A form an orthonormal basis for R", then A is invertible and AT= A-1 . Claim: If the columns of A form...
Problem 2 Let A be an n x n matrix which is not 0 but A-0...
Problem 2 Let A be an n x n matrix which is not 0 but A-0 Let I be the identity matrix. a) (10 Points) Show that A is not diagonalizable. b) (5 Points) Show that A is not invertible. e) (5 Points) Show that I-A is invertible and find its inverse.
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F)...
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
a) Let I be the n x n identity matrix and let O be the n × n zero matrix
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
9. An n × n matrix A is called nilpotent if for-one non, negalivew m, we have Ao, If A is a nilpotent matrix prov conider invertible matrix. To prove this tell me what is (1 + AY first the case w...
9. An n × n matrix A is called nilpotent if for-one non, negalivew m, we have Ao, If A is a nilpotent matrix prov conider invertible matrix. To prove this tell me what is (1 + AY first the case where m2 and in this case show th This should help you to see how to prove the general n x n identity matrix). that 1+ As an Hin at (1+A)---A) case. (I is the
9. An n ×...
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP...
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
Let U and V be nxn orthogonal matrices. Explain why UV is an orthogonal matrix. [That...
Let U and V be nxn orthogonal matrices. Explain why UV is an orthogonal matrix. [That is, explain why UV is invertible and its inverse is (UV)'.] Why is UV invertible? O A. Since U and V are nxn matrices, each is invertible by the definition of invertible matrices. The product of two invertible matrices is also invertible. OB. UV is invertible because it is an orthogonal matrix, and all orthogonal matrices are invertible. O c. Since U and V...
Help on Questions 1-3
Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
[1 2 37 1. Is the matrix 1 0 1 invertible? If yes, what is its inverse? [O 2 -1 2. A matrix is called symmetric if At = A. What can you say about the shape of a symmetric matrix? Give an example of a symmetric matrix that is not a zero matrix. 3. A matrix is called anti-symmetric if A= -A. What can you say about the shape of an anti- symmetric matrix? Give an example of an...
7. Claim: Let A be an (n × n) (square) matrix. ·Claim: If A s invertible and AT = A-1 , then the columns of A form an orthonormal basis for R . Claim: If the columns of A form an orthogonal basis for Rn, then A is invertible and A A-1 . Claim: If the columns of A form an orthonormal basis for R", then A is invertible and AT= A-1 . Claim: If the columns of A form...
Problem 2 Let A be an n x n matrix which is not 0 but A-0 Let I be the identity matrix. a) (10 Points) Show that A is not diagonalizable. b) (5 Points) Show that A is not invertible. e) (5 Points) Show that I-A is invertible and find its inverse.
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
9. An n × n matrix A is called nilpotent if for-one non, negalivew m, we have Ao, If A is a nilpotent matrix prov conider invertible matrix. To prove this tell me what is (1 + AY first the case where m2 and in this case show th This should help you to see how to prove the general n x n identity matrix). that 1+ As an Hin at (1+A)---A) case. (I is the
9. An n ×...
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
Let U and V be nxn orthogonal matrices. Explain why UV is an orthogonal matrix. [That is, explain why UV is invertible and its inverse is (UV)'.] Why is UV invertible? O A. Since U and V are nxn matrices, each is invertible by the definition of invertible matrices. The product of two invertible matrices is also invertible. OB. UV is invertible because it is an orthogonal matrix, and all orthogonal matrices are invertible. O c. Since U and V...
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